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Journal of Mathematical Physics

Rotational diffusions as seen by relativistic observers

Piotr Garbaczewski

Published Oct 1, 1992• 3 citations
Publisher

Abstract

The major unsolved problem in the framework of Nelson’s stochastic mechanics is addressed and an attempt is made to provide a description of relativistic spin-1/2 particles in terms of Markovian diffusions on S3. Random rotations are here labeled by the proper time of a particle in relativistic motion and are continuously distributed along a space-time trajectory followed by the particle in Minkowski space.

DOI
10.1063/1.529887

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The major unsolved problem in the framework of Nelson’s stochastic mechanics is addressed and an attempt is made to provide a description of relativistic spin-1/2 particles in terms of Markovian diffusions on S3. Random rotations are here labeled by the proper time of a particle in relativistic motion and are continuously distributed along a space-time trajectory followed by the particle in Minkowski space.
Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelson's stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.
There are two stochastic descriptions of relativistic quantum spinless particles. In the first one, suggested by Feynman's path integral approach to Klein-Gordon propagator, the main ingredients are diffusions in Minkowski space parametrized by some kind of proper time. The second description, on the contrary, is based upon jumb Markov processes giving the space position of the particle in the sense of Newton and Wigner with the time as parameter. In this paper we bridge the gap between these different probabilistic scenarios by constructing the space jump processes by means of space-time diffusions. We give some aplications to the nonrelativistic limit of the hamiltonian semigroup t→exp−(t/¬h)H where H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 2 +V(.) Il y a deux descriptions stochastiques de la mecanique quantique relativiste pour une particule sans spin. Dans la premiere, suggere par la theorie de Feynman, les ingredients principaux sont des diffusions dans l'espace de Minkowski parametrise par une sorte de temps propre. La deuxieme description, par contre, emploie des processus de Markov a sauts donnant la position de la particule dans l'espace au sens de Newton et Wigner avec le temps comme parametre. Notre papier va relier les deux prescriptions probabilistes car nous construisons les processus a sauts dans l'espace par les diffusions dans l'espace-temps. Nous donnons aussi des applications a l'etude de la limite non relativiste du semi-groupe hamiltonien t→exp−(t/¬h)H ou H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 2 +V(.)
Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.